[Paper Review] Waiter-Client and Client-Waiter Hamiltonicity games on random graphs

Dan Hefetz, Michael Krivelevich+1 more|arXiv (Cornell University)|Sep 17, 2015

Limits and Structures in Graph Theory30 references5 citations

TL;DR

This paper determines the sharp threshold probabilities for Waiter-Client and Client-Waiter Hamiltonicity games on random graphs $\mathcal{G}(n,p)$. For the $(1:q)$ Waiter-Client game, the threshold is $\log n/n$, independent of $q$, while for the Client-Waiter game, it is $(q+1)\log n/n$, showing a dependence on $q$. The results establish the exact edge probabilities at which each player can guarantee a Hamiltonian outcome a.a.s.

ABSTRACT

We study two types of two player, perfect information games with no chance moves, played on the edge set of the binomial random graph ${\mathcal G}(n,p)$. In each round of the $(1 : q)$ Waiter-Client Hamiltonicity game, the first player, called Waiter, offers the second player, called Client, $q+1$ edges of ${\mathcal G}(n,p)$ which have not been offered previously. Client then chooses one of these edges, which he claims, and the remaining $q$ edges go back to Waiter. Waiter wins this game if by the time every edge of ${\mathcal G}(n,p)$ has been claimed by some player, the graph consisting of Client's edges is Hamiltonian; otherwise Client is the winner. Client-Waiter games are defined analogously, the main difference being that Client wins the game if his graph is Hamiltonian and Waiter wins otherwise. In this paper we determine a sharp threshold for both games. Namely, for every fixed positive integer $q$, we prove that the smallest edge probability $p$ for which a.a.s. Waiter has a winning strategy for the $(1 : q)$ Waiter-Client Hamiltonicity game is $(1 + o(1)) \log n/n$, and the smallest $p$ for which a.a.s. Client has a winning strategy for the $(1 : q)$ Client-Waiter Hamiltonicity game is $(q + 1 + o(1)) \log n/n$.

Motivation & Objective

• To determine the exact threshold probability $p$ for which Waiter can guarantee a Hamiltonian outcome in the $(1:q)$ Waiter-Client Hamiltonicity game on $\mathcal{G}(n,p)$.
• To identify the threshold probability $p$ for which Client can guarantee a Hamiltonian outcome in the $(1:q)$ Client-Waiter Hamiltonicity game on $\mathcal{G}(n,p)$.
• To analyze the dependence of these thresholds on the bias parameter $q$ and compare them to the classical threshold for Hamiltonicity in $\mathcal{G}(n,p)$.
• To establish that the Waiter-Client threshold matches the classical Hamiltonicity threshold, while the Client-Waiter threshold scales linearly with $q$.
• To extend the understanding of positional games on random graphs by proving sharp thresholds for two new game types: Waiter-Client and Client-Waiter.

Proposed method

• Use of probabilistic methods and concentration inequalities to analyze the structure of $\mathcal{G}(n,p)$ at critical edge probabilities.
• Construction of a strategy for Waiter to isolate a vertex of low degree in Client’s graph by leveraging the box game framework and independent sets.
• Application of the $(1:q)$ box game strategy to force Client into claiming a vertex with degree one in his graph, thereby preventing Hamiltonicity.
• Use of expectation and concentration bounds to show that, for $p = (q+1 - \varepsilon)\log n / n$, a.a.s. Waiter can isolate a vertex in Client’s graph.
• Establishment of a lower bound on the number of vertices of degree $k$ in $\mathcal{G}(n,p)$, followed by a lower bound on the size of a maximum independent set among such vertices.
• Proof via contradiction: assuming $p$ is below the threshold, showing that Waiter cannot force Client into a non-Hamiltonian graph, thus proving the threshold is sharp.

Experimental results

Research questions

• RQ1 What is the sharp threshold probability $p$ for which Waiter has a winning strategy in the $(1:q)$ Waiter-Client Hamiltonicity game on $\mathcal{G}(n,p)$?
• RQ2 What is the sharp threshold probability $p$ for which Client has a winning strategy in the $(1:q)$ Client-Waiter Hamiltonicity game on $\mathcal{G}(n,p)$?
• RQ3 How does the threshold for the Client-Waiter game depend on the bias parameter $q$?
• RQ4 Why does the Waiter-Client threshold not depend on $q$, while the Client-Waiter threshold does?
• RQ5 Can the threshold probabilities be extended to non-constant $q$, such as $q = O(n)$ or $q \leq (1-o(1))n/\log n$?

Key findings

• For every fixed $q$, the sharp threshold for the $(1:q)$ Waiter-Client Hamiltonicity game on $\mathcal{G}(n,p)$ is $p = (1+o(1))\log n / n$, independent of $q$.
• For every fixed $q$, the sharp threshold for the $(1:q)$ Client-Waiter Hamiltonicity game on $\mathcal{G}(n,p)$ is $p = (q+1+o(1))\log n / n$, which grows linearly with $q$.
• The Waiter-Client threshold matches the classical threshold for Hamiltonicity in $\mathcal{G}(n,p)$, indicating that Waiter can win as soon as a Hamilton cycle is likely to exist.
• The Client-Waiter threshold is strictly larger than the classical threshold, reflecting the increased difficulty for Client to claim a Hamiltonian cycle under adversarial offering.
• A.a.s., for $p = (q+1 - \varepsilon)\log n / n$, Waiter can isolate a vertex in Client’s graph by exploiting low-degree vertices and independent sets, thereby preventing Hamiltonicity.
• The analysis shows that the number of vertices of degree $k$ in $\mathcal{G}(n,p)$ is sufficiently large and concentrated to allow Waiter to construct a strategy that forces Client into a non-Hamiltonian graph below the threshold.